A central aspect of statistical
science is the assessment of dependence among stochastic
variables. The familiar concepts of correlation, regression, and
prediction are special cases, and identification of causal
relationships ultimately rests on representations of multivariate
dependence.
* Graphical Markov models * (GMM) use * graphs*, either
undirected, directed, or mixed, to represent multivariate dependences in
a visual and computationally efficient manner. A GMM is usually
constructed by specifying * local * dependences for each variable, equivalently,
node of the graph in terms of its immediate neighbors and/or
parents by means of undirected and/or directed edges. This simple local
specification can represent a highly varied and complex system of
multivariate dependences by means of the * global * structure of the
graph, thereby obtaining efficiency in modeling, inference, and
probabilistic calculations.
For a fixed graph, equivalently model, the classical methods of
statistical inference may be utilized. In many applied domains, however,
such as expert systems for medical diagnosis or weather forecasting,
or the analysis of gene-expression data, the graph is unknown and is
itself the first goal of the analysis. This poses numerous challenges,
including the following:

- The numbers of possible graphs and models grow superexponentially in the number of variables.
- Distinct
graphs
*G*may be*Markov equivalent*= statistically indistinguishable. - Conversely, the same graph may possess different Markov interpretations.

* Fig. 1. Structure of a chain graph *

- fully characterize Markov equivalence classes of AMP chain graphs via their essential graphs;
- devise efficient methods for model selection and testing;
- develop methods of inference for models comprising submodels of varying dimensionalities;

Alternative Markov properties for chain graphs. Andersson, S., Madigan, D., and Perlman, M. Scandinavian Journal of Statistics 28, 33-85, 2001.

Conditional independence models for seemingly unrelated regressions with incomplete data. Drton, M., Andersson, S., and Perlman, M. Tech. Report No. 431, Dept. of Statistics, University of Washington, Seattle, 2003.

On the bias and mean-square error of order-restricted maximum likelihood estimators. Chaudhuri, S. and Perlman, M. D. Journal of Statistical Planning and Inference, to appear. 2004.

A SINful approach to model selection for gaussian concentration graphs Drton, M. and Perlman, M. Tech. Report No. 429, Dept. of Statistics, University of Washington, Seattle. Biometrika, to appear, 2004.

Pathwise separation and completeness for AMP chain graph Markov models. Annals of Statistics, Levitz, M., Perlman, M. D., and Madigan, D. 29, 1751-1784, 2001.

* Fig. 2. (i) a simple DAG model; (ii) the ancestral graph resulting from marginalizing over t;
(iii) the ancestral graph resulting from conditioning on t *

An alternative approach uses a richer class of graphs, called
* ancestral graphs *. These graphs directly represent the
independence structure among the observed variables that may be
induced by hidden variables. Additional motivation for this class
derives from the causal interpretation of DAG models.

* Fig. 3. (i) a DAG model; (ii) conditioning on s; (iii) marginalizing over l1 and l2;
(iv) conditioning on s and marginalizing over l1 and l2.*

* Fig. 4. Multimodal likelihood associated with the Gaussian ancestral graph in
Fig. 2(ii)*

**Current research projects:**

- Extend the Gaussian fitting algorithms to the discrete data case;
- Analyze likelihoods associated with ancestral graph models; it is known that in some situations the likelihood can be multi-modal. See Figure
- Graphical analysis of non-independence constraints associated with hidden variable models;
- Characterization of Markov equivalence classes of ancestral graph models;
- Graphical analysis of the strength of dependence in DAG models.

Ancestral graph Markov models Thomas Richardson and Peter Spirtes. Annals of Statistics, 2002.

Multimodality of the likelihood in the bivariate seemingly unrelated regression model Mathias Drton and Thomas Richardson. To appear in Biometrika, 2004

Causal inference via ancestral graph Markov models (with discussion). Thomas Richardson and Peter Spirtes. In
* Highly Structured Stochastic Systems * P.J. Green, N.L. Hjort and S. Richardson (eds.).

A new algorithm for maximum likelihood estimation in Gaussian graphical models for marginal independence. Mathias Drton and Thomas Richardson. Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence, 2003

Iterative Conditional Fitting for Gaussian Ancestral Graph Models Mathias Drton and Thomas Richardson. Department of Statistics Technical Report No. 437

Using the structure of d-connecting paths as a qualitative measure of the strength of dependence. Sanjay Chaudhuri and Thomas Richardson. Proceedings of the Nineteenth Conference on Uncertainty in Artificial Intelligence, 2003

Markov Equivalence Classes for Maximal Ancestral Graphs Ayesha Ali and Thomas Richardson, Proceedings of the Eighteenth Conference on Uncertainty in Artificial Intelligence, 2002.

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